Milwaukee trailed most of the night but Bogut took control in the second half with 13 points. Kelenna Azubuike led the Warriors with 24 points while Corey Maggette added 16.
Bucks coach Scott Skiles said he changed his strategy in the second half because of a mismatch.
"We were going to Bogut a lot because they had a small player on him," he said.
With seconds to play, Azubuike missed a 3-pointer that would have tied it at 95. The Bucks then hit free throws to seal the victory.
Bucks rookie Joe Alexander, who finished with 11 points, was the crowd favorite in this southern Chinese city. He's the first NBA player to be born in Taiwan and can speak Mandarin.
The Warriors took a 75-70 lead going into the last period. But the Bucks tied it at 85 with about six minutes to play when Alexander hit a short turnaround from the baseline.
The Bucks broke to a 10-3 lead, with Richard Jefferson making two 3-pointers in the first three minutes. But Milwaukee went scoreless for nearly four minutes, and the Warriors finished the quarter ahead 27-18. Golden State was led by Azubuike, who came off the bench to score seven points during the Bucks' drought.
In the second quarter, Milwaukee drew within two points of the Warriors as center Francisco Elson scored six points to narrow the gap at 31-33. Anthony Morrow, who didn't play in the first quarter, gave the Warriors a spark, scoring six points to put them up 48-41.
Alexander, who played for West Virginia, was the Bucks' top pick in the last NBA draft, and Skiles said he has a lot to learn.
"He doesn't fully understand yet what we're trying to do out there, but he has a lot of energy and when he does understand what we're trying to do, he'll be a good player," Skiles said.
Maggette said the Warriors need more work before the season starts.
"I think our team was a little slow," he said. "We also have to get better on the defensive end."